Approximation order and approximate sum rules in subdivision

Abstract

Several properties of stationary subdivision schemes are nowadays well understood. In particular, it is known that the polynomial generation and reproduction capability of a stationary subdivision scheme is strongly connected with sum rules, its convergence, smoothness and approximation order. The aim of this paper is to show that, in the non-stationary case, exponential polynomials and approximate sum rules play an analogous role of polynomials and sum rules in the stationary case. Indeed, in the non-stationary univariate case we are able to show the following important facts: i) reproduction of N exponential polynomials implies approximate sum rules of order N ; ii) generation of N exponential polynomials implies approximate sum rules of order N , under the additional assumption of asymptotical similarity and reproduction of one exponential polynomial; iii) reproduction of an N -dimensional space of exponential polynomials and asymptotical similarity imply approximation order N ; iv) the sequence of basic limit functions of a nonstationary scheme reproducing one exponential polynomial converges uniformly to the basic limit function of the asymptotically similar stationary scheme.

Exponential pseudo-splines: Looking beyond exponential B-splines

“…The proof of (6.5) is based on the recent results proven in [18,Theorem 10] and is a direct consequence of the exponential polynomial generation properties of {a (k) M,N,Γ (z), k ≥ 0} and the asymptotical similarity of {a (k) M,N,Γ , k ≥ 0} to {a M,N }, previously shown in Corollary 6.5. Then, for the convergence and regularity result we can rely on [6].…”

“…We recall that, while the term generation usually refers to the subdivision scheme capability of providing specific types of limit functions, with reproduction we mean the capability of a subdivision scheme to reproduce in the limit exactly the same function from which the data are sampled. The property of reproduction of exponential polynomials is also important since strictly connected to the approximation order of subdivision schemes and to their regularity (see [18]). In fact, the higher is the number of exponential polynomials reproduced, the higher is the approximation order and the possible regularity of the scheme.…”

“…Finally, we additionally provide a convergence and regularity analysis of the nonstationary subdivision schemes corresponding to the exponential pseudo-spline symbols here derived. This is possible by first showing that exponential pseudo-spline schemes are asymptotically similar to polynomial pseudo-spline schemes, and then combining recent advances on convergence and regularity of nonstationary subdivision schemes presented in [6] and in [18].…”

A Family of Binary Univariate Nonstationary Quasi‐Interpolatory Subdivision Reproducing Exponential Polynomials

“…From eorem 4 and Remark 1, the new scheme S a k ΓΛ n ,ω k k≥0 reproduces EP ΓΛ n and generates EP ΓΛ n+1 with ΓΛ n ⊂ ΓΛ n+1 . en, according to [36], Corollary 14, this new nonstationary scheme satisfies approximate sum rules of order r with r � 2τ 0 + 2 l j�1 τ l � 2[(n + 1) + 1] � 2n + 4. en, from eorem 1, this new nonstationary scheme is C r convergent, if the free parameter ω k is such that the asymptotical similar subdivision is C r convergent with r < 2n + 4. □ Now, let us discuss the approximation order of the scheme S a k ΓΛ n ,ω k k≥0 .…”

Level-Dependent Interpolatory Hermite Subdivision Schemes and Wavelets

“…[3,8,9,14,19]. As in standard (non-Hermite) schemes, such a property is crucial for assuring not only the convergence of the scheme, but also the smoothness and the approximation order of its limit function [6,12,15].…”